Stochastic Galerkin Matrices

Ernst, Oliver G.; Ullmann, Elisabeth

doi:10.1137/080742282

Cited by 74 publications

(89 citation statements)

References 18 publications

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“…Unfortunately, explicit formulae for their eigenvalues remain elusive (except for low values of d, see [12]). It can be shown, however, that they are very illconditioned with respect to the discretization parameter d. In Corollary 4.6 we will show the spectral radius of each one can be bounded above by a quantity that is O(exp(M d) exp(|α|/2)).…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…The eigenvalues of the matrices G n in (3.7a) are known explicitly (see [27], [12]). For Gaussian random variables, the condition number of each G n grows, at worst, like O( √ d).…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…To this end, we recall, first, an eigenvalue bound from [12]. Denote by (η m,i , w m,i ) δm i=1 the nodes and weights, respectively, of the δ m -point Gaussian quadrature rule associated with ρ m in (2.15).…”

Section: Schur Complement Preconditioners Applyingmentioning

confidence: 99%

“…, G N represent multiplication operators on a probability space associated with the random PDE coefficients. Their structural and spectral properties (see [12], [27], [30]) are governed by our choice of discretization on the probability space. We assume that the (1,1) block in (1.4) is positive definite and so linear systems with this C can be solved via preconditioned minres with the block-diagonal preconditioners described above.…”

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confidence: 99%

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Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial.We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

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Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

“…The eigenvalues of the matrices G n in (3.7a) are known explicitly (see [27], [12]). For Gaussian random variables, the condition number of each G n grows, at worst, like O( √ d).…”

Section: Lognormal Diffusion Coefficient the Question Remains As To mentioning

confidence: 99%

Section: Schur Complement Preconditioners Applyingmentioning

confidence: 99%

mentioning

confidence: 99%

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Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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“…The matrices D i , G 2 ∈ R Q×Q characterize the stochastics of the problem and are respectively defined by D i = Ψ i ΨΨ T and G 2 = ξ 2 ΨΨ T . In [26,10] properties of these matrices are given. Each individual matrix is a sparse matrix, however the sum, …”

Section: Newton Linearizationmentioning

confidence: 99%

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Rosseel

Gersem

Vandewalle

2010

CiCP

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The stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost. AbstractThe stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost.

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Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Sousedík

Ghanem

Phipps

2013

Numer. Linear Algebra Appl.

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Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Loève expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments.HIERARCHICAL SCHUR COMPLEMENT PRECONDITIONER 137 Both methods are defined using tensor product spaces for the spatial and stochastic discretizations. Collocation methods sample the stochastic PDE at a set of collocation points, which yields a set of mutually independent deterministic problems. Because one can use existing software to solve this set of problems, collocation methods are often referred to as non-intrusive. However, the number of collocation points can be quite prohibitive when high accuracy is required or when the stochastic problem is described by a large number of random variables.On the other hand, the stochastic Galerkin method is intrusive. It uses the spectral finite element approach to transform a stochastic PDE into a coupled set of deterministic PDEs, and because of this coupling, specialized solvers are required. The design of iterative solvers for systems of linear algebraic equations obtained from discretizations by stochastic Galerkin finite element methods has received significant attention recently. It is well-known that suitable preconditioning can significantly improve convergence of Krylov subspace iterative methods. Among the most simple, yet quite powerful methods, belongs the mean-based preconditioner by Powell and Elman [6], see also [7]. Further improvements include, for example, the Kronecker product preconditioner by Ullmann [8]. We refer to Rosseel and Vandewalle [9] for a more complete overview and comparison of various iterative methods and preconditioners, including matrix splitting and multigrid techniques. Also, an intere...

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Stochastic Galerkin Matrices

Cited by 74 publications

References 18 publications

Preconditioning Stochastic Galerkin Saddle Point Systems

Preconditioning Stochastic Galerkin Saddle Point Systems

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

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